Download Spin current and rectification in one-dimensional electronic systems Bernd Braunecker,

PHYSICAL REVIEW B 76, 085119 共2007兲
Spin current and rectification in one-dimensional electronic systems
Bernd Braunecker,1,2 D. E. Feldman,1 and Feifei Li1
1Department
2Department
of Physics, Brown University, Providence, Rhode Island 02912, USA
of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
共Received 19 June 2007; published 16 August 2007兲
We demonstrate that spin current can be generated by an ac voltage in a one-channel quantum wire with
strong repulsive electron interactions in the presence of a nonmagnetic impurity and uniform static magnetic
field. In a certain range of voltages, the spin current can exhibit a power dependence on the ac voltage bias with
a negative exponent. The spin current expressed in units of ប / 2 per second can become much larger than the
charge current in units of the electron charge per second. The spin current generation requires neither spinpolarized particle injection nor time-dependent magnetic fields.
DOI: 10.1103/PhysRevB.76.085119
PACS number共s兲: 73.63.Nm, 71.10.Pm, 73.40.Ei
I. INTRODUCTION
The pioneering paper by Christen and Büttiker1 has stimulated much interest to rectification in quantum wires and
other mesoscopic systems. Most attention was focused on the
simplest case of Fermi liquids.2,3 Recently, this research was
extended to strongly interacting systems where Luttinger liquids are formed.4–7 One of the topics of current interest is the
rectification effect in Luttinger liquids in a magnetic field.3,6,7
In the presence of a magnetic field, both spin and charge
currents can be generated. So far, however, only charge currents in Luttinger liquids have been studied. In this paper, we
show that a dc spin current can be generated by an ac voltage
bias in a single-channel quantum wire.
In recent years, many approaches to the generation of spin
currents in quantum wires were put forward. Typically, both
spin and charge currents are generated, and the spin current
expressed in units of ប / 2 per second is smaller than the
electric current in units of e per second 共e is the electron
charge兲. Such a situation naturally emerges in partially polarized systems, since each electron carries the charge e and
its spin projection on the z axis is ±ប / 2. A proposal on how
to obtain a spin current exceeding the charge current in a
quantum wire was published by Sharma and Chamon8 who
considered a Luttinger liquid in the presence of a timedependent magnetic field in a region of the size of an electron wavelength. In a very different physical context, a spin
current without charge current was predicted for edge modes
in the quantum Hall effect in graphene.9 Pure spin currents
can also flow in open circuits, which cannot support charge
currents.10 In this paper, we show that the generation of a dc
spin current exceeding the charge current is also possible in
closed circuits without time-dependent magnetic fields. The
spin current can be generated in a spatially asymmetric Luttinger liquid system in the presence of an ac bias. Interestingly, in a certain interval of low voltages, the dc spin current
grows as a negative power of the ac voltage when the voltage
decreases.
The paper is organized as follows. The next section contains a qualitative discussion. We briefly address the simplest
case of noninteracting electrons, discuss its differences from
the most interesting case of strong electron interaction, and
estimate at what conditions the effect can be observed. Sec1098-0121/2007/76共8兲/085119共11兲
tion III contains the details of the bosonization procedure
which we use to treat the electron-electron interaction. In
Sec. IV, we calculate analytically the spin and charge rectification currents in the presence of a weak asymmetric potential. Numerical results for a simple model with strong
asymmetric potential are discussed in Appendix A. Appendices B and C contain technical details of the perturbation
theory employed in Sec. IV.
II. MODEL AND PHYSICS OF THE PROBLEM
The rectifying quantum wire is sketched in Fig. 1. It consists of a one-dimensional conductor with a scatterer in the
center of the system at x = 0. The scatterer creates an asymmetric potential U共x兲 ⫽ U共−x兲. The size of the scatterer aU
⬃ 1 / kF is of the order of the electron wavelength. A spin
current can be generated only if time-reversal symmetry is
broken. Thus, we assume that the system is placed in a uniform magnetic field H. The field defines the Sz direction of
the electron spins. If the wire is sufficiently narrow, then the
effect of the magnetic field on the kinetic energy of electrons
can be neglected and the field enters the problem only via its
interaction with the spins. At its two ends, the wire is connected to nonmagnetic electrodes, labeled by i = 1 , 2. The left
electrode, i = 1, is controlled by an ac voltage source, while
the right electrode, i = 2, is kept on ground.
The magnetic field H breaks the symmetry between the
two orientations of the electron spin. In a uniform wire, this
would not result in a net spin current since the conductances
quantum wire
±V
contact
i=1
H
U (x)
aU
contact
i=2
FIG. 1. Sketch of the one-dimensional conductor connected to
two electrodes on both ends. Currents are driven through a voltage
bias V that is applied on the left electrode while the right electrode
is kept on ground. The system is magnetized by the field H. Electrons are backscattered off the asymmetric potential U共x兲. U共x兲
⫽ 0 in the region of size aU ⬃ 1 / kF.
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PHYSICAL REVIEW B 76, 085119 共2007兲
BRAUNECKER, FELDMAN, AND LI
I共␮R,Sz兲 =
FIG. 2. Double-well potential with quasistationary levels. The
transmission coefficient is maximal in the shaded regions. The narrow potentials u1共x兲 and u2共x兲 are centered at the positions x = 0 and
x = a 共a ⬍ k−1
F 兲, respectively, and are modeled by ␦ functions in Eq.
共A1兲.
of the spin-up and -down channels would be the same,17
e2 / h, and the spin currents of the spin-up and -down electrons would be opposite. In the presence of a potential barrier, such a cancellation does not occur.18 In a system with
strong electron interaction, the spatial asymmetry of the wire
leads to an asymmetric I-V curve,4 I共V兲 ⫽ I共−V兲. Thus, an ac
voltage bias generates spin and charge dc currents, Isr and Irc.
As we will see, the problem is most interesting in the case
of strong electron interaction. Before addressing that more
difficult case, let us discuss what happens in the absence of
electron interaction. By noninteracting system, we mean a
wire in which electron-electron interaction is completely
screened by the gates. In such situation, the charge density in
the wire is not fixed but depends on the gate potential and the
electrochemical potentials of the leads. The leads define the
chemical potentials ␮L and ␮R of the left- and right-moving
electrons, which are injected from the right and left reservoirs, respectively. In what follows, we will assume that the
chemical potential ␮L = 0 and ␮R oscillates between +eV and
−eV. Thus, in the presence of the magnetic field H, the Fermi
energies counted from the band bottom equal EFL共Sz兲 = EF
+ 2Sz␮H for the left-moving electrons and EFR共Sz兲 = EF + ␮R
+ 2Sz␮H for the right-moving electron, where Sz = ± 1 / 2 is
the electron spin projection, EF the Fermi level in the absence of the magnetic field and voltage bias, and ␮ the electron magnetic moment.
As we will see, in a strongly interacting system, the form
of the potential barrier U共x兲 plays little role. However, it is
crucial in the noninteracting case. Let us choose U共x兲 in the
form of the double potential barrier so that quasistationary
levels En, n = 0 , 1 , . . ., are present 共Fig. 2兲. Thus, in the noninteracting case, we consider a resonant tunneling diode.11
The spin and charge currents as functions of the chemical
potential ␮R are
Ic共␮R兲 = I共␮R,1/2兲 + I共␮R,− 1/2兲,
I s共 ␮ R兲 =
ប
关I共␮R,1/2兲 − I共␮R,− 1/2兲兴,
2e
where e is the electron charge,
共1兲
共2兲
e
h
冕
R
EF
共Sz兲
L
EF
共Sz兲
dET共E兲,
共3兲
and T共E兲 is the transmission coefficient. The transmission
coefficient is small far from the energies of the quasistationary levels E = En and increases as E approaches En. Obviously, if one applies a dc voltage eV = ␮R, then the charge
current, expressed in units of e per second, exceeds the spin
current in units of ប / 2 per second. The situation changes in
the presence of an ac bias. The dc currents generated by
r
= 关Ic/s共V兲
an ac voltage bias can be estimated as Ic/s
+ Ic/s共−V兲兴 / 2. Let us now assume that the magnetic field is
tuned such that the Fermi level of the spin-down electrons
EF − ␮H is close to the quasistationary level E0 and exceeds
E0, while the Fermi level of the spin-up electrons EF + ␮H is
close to E1 and lies below E1. Also, let eV be smaller than
the distances 兩E0 − EF + ␮H兩 and 兩E1 − EF − ␮H兩 between the
Fermi levels and quasistationary levels in the absence of the
voltage bias. In addition, we assume that T共EF + ␮H兲 = T共EF
− ␮H兲. From the energy dependence of the transmission coefficient T共E兲 near the resonant levels, one finds that
I共eV , 1 / 2兲 ⬎ I共eV , −1 / 2兲 and 兩I共−eV , 1 / 2兲兩 ⬍ 兩I共−eV , −1 / 2兲兩.
ប r
Ic. By an appropriate choice of parameters, one
Hence, Isr ⬎ 2e
can produce any ratio of the spin and charge rectification
currents.
Note that the rectification effect for noninteracting electrons is possible even if the potential U共x兲 is symmetric. The
asymmetry of the system, necessary for rectification, is introduced by the applied voltage bias. The charge density injected into the wire from the leads is proportional to ␮L
+ ␮R and hence is different for the opposite signs of the voltage. If the injected charge density were independent of the
voltage sign, i.e., ␮R oscillated between eV / 2 and −eV / 2 and
␮L = −␮R oscillated between −eV / 2 and eV / 2, then the rectification effect would be impossible for noninteracting electrons. This follows from Eqs. 共1兲–共3兲 and the fact that the
transmission coefficient T共E兲 is independent of the direction
of the incoming wave for noninteracting particles.12 In the
presence of electron repulsion, both the asymmetry of the
potential and the voltage dependence of the injected charge
contribute to the rectification current. It turns out that in the
case of strong electron interaction, the rectification effect due
to the asymmetry of the potential barrier dominates.
The above example is based on a special form of the
potential barrier in the wire and assumes that the magnetic
field and chemical potentials are tuned in order to obtain the
desired effect. As shown below, in the presence of strong
repulsive electron interaction, no tuning is necessary and no
quasistationary states are needed to obtain the spin current
which is greater than the charge current. In fact, the spin
rectification effect is possible even for weak asymmetric potentials U共x兲. This can be understood from the following toy
model 共a related model for rectification in a two-dimensional
electron gas was studied in Ref. 13兲: Let there be no uniform
magnetic field H and no asymmetric potential U共x兲. Instead,
both right↔ left and spin-up↔ spin-down symmetries are
broken by a weak coordinate-dependent magnetic field
Bz共x兲 ⫽ Bz共−x兲, which is localized in a small region of size
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SPIN CURRENT AND RECTIFICATION IN ONE-…
FIG. 3. Normalized charge rectification current Ic / Ic0 and spin
rectification current Is / Is0 versus applied voltage V / V0 for noninteracting electrons with EF = 400⑀0, ␮H = 75⑀0, u1 = 50⑀0a, and u2 =
−50⑀0a, where ⑀0 = ប2 / ma2 共see Fig. 2 and Appendix A兲. Ic0
= 50e⑀0 / ប, Is0 = 25⑀0, and V0 = 50⑀0 / e are arbitrary reference currents and voltage.
FIG. 4. Normalized charge rectification current Ic / Ic0 and spin
rectification current Is / Is0 versus applied voltage V / V0 for interacting electrons with EF = 100⑀0, ␮H = 25⑀0, ␥ = 12.6⑀0a / e, u1 = 25⑀0a,
and u2 = 50⑀0a, where ⑀0 = ប2 / ma2 共see Fig. 2 and Appendix A兲.
Ic0 = 50e⑀0 / ប, Is0 = 25⑀0, and V0 = 50⑀0 / e are arbitrary reference currents and voltage.
⬃1 / kF 共we do not include the components Bx,y in the toy
model兲. Let us also assume that the spin-up and -down electrons do not interact with the electrons of the opposite spin.
Then, the system can be described as the combination of two
spin-polarized one-channel wires with opposite spindependent potentials ±␮Bz共x兲, where ␮ is the electron magnetic moment. According to Ref. 4, an ac bias generates a
rectification current in each of those two systems and the
currents are proportional to the cubes of the potentials
共±␮Bz兲3. Thus, Ir↑ = −Ir↓. Hence, no net charge current Ir = Ir↑
+ Ir↓ is generated in the leading order. At the same time, there
is a nonzero spin current in the third order in Bz. A similar
effect is present in a more realistic Luttinger liquid model
considered below.
The main focus of this paper is on the case of weak asymmetric potentials. A simple model with strong impurities is
studied in Appendix A. In Figs. 3 and 4, we have represented
the results from a numerical evaluation for the spin and
r
for the potential shown in Fig. 2. Figure 3
charge currents Is,c
shows the noninteracting case. In Fig. 4, we represent the
case of strong electron interaction. We have chosen parameters 共explained in the figure captions兲 such that Irc is smaller
than Isr for a range of the applied voltage. Further information
on the numerical approach is given in Appendix A.
Transport in a strongly interacting system in the presence
of a strong asymmetric potential U共x兲 is a difficult problem
which cannot be solved analytically and is sensitive to a
particular choice of the potential. As we have mentioned,
Appendix A contains the numerical analysis of a simple
model of interacting electrons with a strong potential barrier.
On the other hand, the interacting problem can be solved
analytically in the limit of a weak potential U共x兲 with the
help of the bosonization and Keldysh techniques 共Sec. III兲.
We will see that the rectification current exhibits a number of
universal features, independent of the form of the potential
U共x兲. In particular, in a wide interval of interaction strength,
the spin rectification current can exceed the charge rectification current for an arbitrary shape of the asymmetric potential barrier.
Rectification is a nonlinear transport phenomenon. Thus,
it cannot be observed at low voltages at which the I-V curve
is linear and hence symmetric. In Luttinger liquids, the I-V
curve is nonlinear at eV ⬎ kBT, where T is the temperature.14
We will concentrate on the limit of the zero temperature
which corresponds to the strongest rectification. We expect
qualitatively the same behavior at T ⬃ V. At higher temperatures, the charge and spin rectification effects disappear.
Since the temperatures of the order of millikelvins can be
achieved with dilution refrigeration, the rectification effect is
possible even for the voltages as low as V ⱗ 1␮V.
In this paper, we focus on the low-frequency ac bias. We
define the rectification current as the dc response to a lowfrequency square voltage wave of amplitude V:
Isr共V兲 = 关Is共V兲 + Is共− V兲兴/2,
共4兲
Irc共V兲 = 关Ic共V兲 + Ic共− V兲兴/2.
共5兲
The above dc currents are expressed via the currents of
spin-up and -down electrons: Irc = Ir↑ + Ir↓, Isr = 共ប / 2e兲关Ir↑ − Ir↓兴.
The spin current exceeds the charge current if the signs of Ir↑
and Ir↓ are opposite. Equations 共4兲 and 共5兲 for the dc do not
contain the frequency ␻ of the ac bias. They are valid as long
as the frequency
␻ ⬍ eV/ប.
共6兲
Indeed, as shown below, the rectification current is determined by electron backscattering off the asymmetric poten-
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BRAUNECKER, FELDMAN, AND LI
tial. Hence, one can neglect the time dependence of the ac
voltage in Eqs. 共4兲 and 共5兲 if the period of the ac bias exceeds
the duration ␶ of one backscattering event. The time ␶
⬃ ␶travel + ␶uncertainty includes two contributions. ␶travel is the
time of the electron travel across the potential barrier.
␶uncertainty comes from the uncertainty of the energy of the
backscattered particle. If the barrier amplitude U共x兲 ⬍ EF and
the barrier occupies a region of size aU ⬃ 1 / kF, then ␶travel
⬃ 1 / kFv ⬃ ប / EF, where v ⬃ បkF / m is the electron velocity.
The energy uncertainty ⬃eV translates into ␶uncertainty
⬃ ប / eV. Thus, for eV ⬍ EF, one obtains condition 共6兲. The
same condition can be derived with the approach of Appendix A of Ref. 15 and emerges in a related problem.16 Note
that for realistic voltages, the low-frequency condition 共6兲
allows rather high frequencies. Even for V ⬃ 1 ␮V, the maximal ␻ ⬃ 1 GHz.
There remains the question of the asymmetric impurity:
We require a potential U共x兲 that is localized within ⬃1 / kF. A
possible realization is to generate two different 共symmetric兲
local potentials by two gates within a distance ⬃1 / kF or an
electric potential created by an asymmetric gate of size
⬃1 / kF placed at the distance ⬃1 / kF from the wire. Electron
densities of ␳ ⬃ 1011 cm−2 are possible nowadays in twodimensional electron gases, yielding 1 / kF up to several
10 nm. Confinement in a one-dimensional wire will reduce
the electron density further so that this number may increase
further. Modern techniques allow placing electric gates of
widths of ⬃20 nm at distances of ⬃20– 50 nm. A realization
of an asymmetric potential in this way is, therefore, within
the reach. Alternatively, in the case of shorter electron wavelength, it should be possible to place an asymmetrically
shaped scanning tunneling microscope tip close to the wire.
An applied bias would yield an asymmetric scattering potential. With such a tip, the asymmetry cannot be directly tuned,
but most of our predictions are not sensitive to the precise
shape of the potential. Certainly, an asymmetric potential
may simply emerge by chance due to the presence of two
point impurities of unequal strengths at the distance ⬃1 / kF.
metric potential is weak, U共x兲 ⬍ EF. This will enable us to
use perturbation theory.
We assume that the magnetic field H couples only to the
electron spin, and we neglect the correction −eA / c to the
momentum in the electron kinetic energy. Indeed, for a uniform field, one can choose A ⬃ y, where the y axis is orthogonal to the wire and y is small inside a narrow wire. As
shown in Ref. 18, such a system allows a formulation within
the bosonization language and, in the absence of the asymmetric potential, can be described by a quadratic bosonic
Hamiltonian
H0 =
兺
冕
dx共⳵x␾␯␴兲H␯␴,␯⬘␴⬘共⳵x␾␯⬘␴⬘兲, 共7兲
where ␴ is the spin projection and ␯ = R , L labels the left- and
right-moving electrons, which are related to the boson fields
†
† ±i关kF␯␴x+␾␯␴共x兲兴
␾␯␴ as ␺␯␴
共x兲 ⬃ ␩␯␴
e
with ⫾ for ␯ = R , L. The
†
operators ␩␯␴ are the Klein factors adding a particle of type
共␯ , ␴兲 to the system, and kF␯␴ / ␲ is the density of 共␯ , ␴兲 particles in the system. The densities of the spin-up and -down
electrons are different since the system is polarized by the
external magnetic field. The 4 ⫻ 4 matrix H describes the
electron-electron interactions. In the absence of spin-orbit
interactions, L ↔ R parity is conserved and we can introduce
the quantities ␾␴ = ␾L␴ + ␾R␴ and ⌸␴ = ␾L␴ − ␾R␴ such that the
Hamiltonian decouples into two terms depending on ␾␴ and
⌸␴ only. In the absence of the external field, this Hamiltonian would further be diagonalized by the combinations
␾c,s ⬀ ␾↑ ± ␾↓, and similarly for ⌸c,s, expressing the spin and
charge separation. Here, this is no longer the case because of
the external magnetic field. If we focus on the ␾ fields only
关as ⌸ will not appear in the operators describing backscatter˜ c,s,
ing off U共x兲兴, the fields diagonalizing the Hamiltonian, ␾
have a more complicated linear relation to ␾↑,↓, which we
can write as
冉 冊 冉冑冑
␾↑
=
␾↓
III. BOSONIZATION AND KELDYSH TECHNIQUE
At ␻ ⬍ V, the calculation of the rectification currents reduces to the calculation of the stationary contributions to the
dc I-V curves Is共V兲 and Ic共V兲 that are even in the voltage V.
We assume that the Coulomb interaction between distant
charges is screened by the gates. This will allow us to use the
standard Tomonaga-Luttinger model with short range
interactions.14 Electric fields of external charges are also assumed to be screened. Thus, the applied voltage reveals itself
only as the difference of the electrochemical potentials E1
and E2 of the particles injected from the left and right reservoirs. We assume that one lead is connected to the ground so
that its electrochemical potential E2 = EF is fixed. The electrochemical potential of the second lead E1 = EF + eV is controlled by the voltage source 共see Fig. 1兲. Since the
Tomonaga-Luttinger model captures only low-energy physics, we assume that eV ⬍ EF, where EF is of the order of the
bandwidth. Rectification occurs due to backscattering off the
asymmetric potential U共x兲. We will assume that the asym-
兺
␯,␯⬘=L,R ␴,␴⬘=↑,↓
gc关1 + ␣兴
gc关1 − ␣兴
冑gs关1 + ␤兴
− 冑gs关1 − ␤兴
冊冉 冊
˜c
␾
,
˜s
␾
共8兲
and which corresponds to the matrix ÂT of Ref. 18. The
normalization has been chosen such that the propagator of
˜ fields with respect to the Hamiltonian 关Eq. 共7兲兴 is
the ␾
˜ c,s共t1兲␾
˜ c,s共t2兲典 = −2 ln(i共t1 − t2兲 / ␶c + ␦), where
evaluated to 具␾
␦ ⬎ 0 is an infinitesimal quantity and ␶c ⬃ ប / EF the ultraviolet cutoff time. For noninteracting electrons without a magnetic field, gc = gs = 1 / 2. gc ⬍ 1 / 2 共⬎1 / 2兲 for repulsive 共attractive兲 interactions. The interaction constants depend on
microscopic details and the magnetic field. The dimensionless parameter which controls the interaction strength is the
ratio of the potential and kinetic energies of the electrons.
This ratio grows as the charge density decreases, and hence
lower electron densities correspond to stronger repulsive interaction. In the absence of the magnetic field, terms in Eq.
共7兲 in the form of exp共±2i冑gs␾s兲 may become relevant and
open a spin gap for gs ⬍ 1 / 2. In our model, they can be
neglected since they are suppressed by the rapidly oscillating
factors exp共±2i关kF↑ − kF↓兴x兲. It is convenient to model the
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SPIN CURRENT AND RECTIFICATION IN ONE-…
leads as the regions near the right and left ends of the wire
without electron interaction.17
Backscattering off the impurity potential U共x兲 is described by the following contribution to the Hamiltonian14
H = H 0 + H ⬘:
H⬘ =
U共n↑,n↓兲ein ␾ 共0兲+in ␾ 共0兲 ,
兺
n ,n
↑ ↑
共9兲
↓ ↓
↑ ↓
where the fields are evaluated at the impurity position x = 0
and U共n↑ , n↓兲 = U*共−n↑ , −n↓兲, since the Hamiltonian is Hermitian. The fields ⌸ do not enter the above equation due to
the conservation of the electric charge and the z projection of
the spin. The Klein factors are not written because they drop
out in the perturbative expansion. U共n↑ , n↓兲 are the amplitudes of backscattering of n↑ spin-up and n↓ spin-down particles with n␴ ⬎ 0 for L → R and n␴ ⬍ 0 for R → L
scattering. U共n↑ , n↓兲 can be estimated as14 U共n↑ , n↓兲
⬃ kF 兰 dxU共x兲ein↑2kF↑x+in↓2kF↓x ⬃ U, where U is the maximum
of U共x兲. In the case of a symmetric potential, U共x兲 = U共−x兲,
the coefficients U共n↑ , n↓兲 are real.
The spin and charge currents can be expressed as
1
1
2
2
+ Rs,c
= Ls,c
+ Rs,c
,
Is,c = Ls,c
i
Ls,c
共10兲
bs
bs
= 具0兩S共− ⬁,0兲Îc,s
S共0,− ⬁兲兩0典,
Ic,s
dQ̂R/dt = i关H,Q̂R兴/ប
=
IV. RECTIFICATION CURRENTS
In order to calculate the current 关Eq. 共13兲兴, we will expand
the evolution operator in powers of U共n , m兲. Such perturbative approach is valid only if U ⬍ EF. The details of the perturbative calculation are discussed in Appendix C.
The currents 关Eq. 共13兲兴 can be estimated using a renormalization group procedure.14 As we change the energy scale
E, the backscattering amplitudes U共n↑ , n↓兲 scale as
U共n,m;E兲 ⬃ U共n,m兲共E/EF兲z共n,m兲 ,
共14兲
where the scaling dimensions are
z共n,m兲 = n2关gc共1 + ␣兲2 + gs共1 + ␤兲2兴 + m2关gc共1 − ␣兲2
+ gs共1 − ␤兲2兴 + 2nm关gc共1 − ␣2兲 − gs共1 − ␤2兲兴 − 1
− ie
兺 共n↑ + n↓兲U共n↑,n↓兲ein↑␾↑共0兲+in↓␾↓共0兲 ,
ប n↑,n↓
= n2A + m2B + 2nmC − 1.
共11兲
Îsbs = dŜR/dt = −
共13兲
where 兩0典 is the ground state for the Hamiltonian H0, Eq. 共7兲,
and S共t , t⬘兲 the evolution operator for H⬘ from t⬘ to t in the
interaction representation with respect to H0. The result of
this calculation depends on the elements of the matrix 关Eq.
共8兲兴, which describe the low-energy degrees of freedom and
depend on the microscopic details. Several regimes are
possible5 at different values of the parameters gs ⬎ 0, gc ⬎ 0,
␣, and ␤. In this paper, we focus on one particular regime in
which the main contribution to the rectification current
comes from backscattering operators U共1 , 0兲, U共0 , −1兲, and
U共−1 , 1兲.
i
Rs,c
and
denote the current of the left and right
where
movers near electrode i, respectively 共see Fig. 2兲. For a clean
1
2
1
2
= Rs,c
, Ls,c
= Ls,c
and
system 关U共x兲 = 0兴, the currents obey17 Rs,c
2
Ic = 2e V / h, Is = 0. With backscattering off U共x兲, particles are
transferred between L and R in the wire, and hence Rs2 = Rs1
+ dSR / dt and R2c = R1c + dQR / dt, where QR and SR denote the
total charge and the z projection of the spin of the right2
1
and Rs,c
are determined
moving electrons.16 The currents Ls,c
by the leads 共i.e., the regions without electron interaction in
our model17兲 and remain the same as in the absence of the
asymmetric potential. Thus, the spin and charge currents can
bs
be represented as Ic = 2e2V / h + Ibs
c and Is = Is , where the backscattering current operators are4,5,14
Îbs
c =
state determines the bare Keldysh Green’s functions.
We will consider only the zero temperature limit. It is
convenient to switch16 to the interaction representation H0
→ H0 − E1NR − E2NL. This transformation induces a time dependence in the electron creation and annihilation operators.
As a result, each exponent in Eq. 共9兲 is multiplied by
exp共ieVt关n↑ + n↓兴 / ប兲.
In the Keldysh formulation,19 the backscattering currents
关Eqs. 共11兲 and 共12兲兴 are evaluated as
i
兺 共n↑ − n↓兲U共n↑,n↓兲ein↑␾↑共0兲+in↓␾↓共0兲 .
2 n↑,n↓
The renormalization group 共RG兲 stops at the scale of the
order E ⬃ eV. At this scale, the backscattering current can be
bs
= Vrc,s共V兲, where the effective reflection
represented as Ic,s
coefficient rc,s共V兲 is given by the sum of contributions of the
form4,14
共12兲
The calculation of the rectification currents reduces to the
calculation of the currents 关Eqs. 共11兲 and 共12兲兴 at two opposite values of the dc voltage.
To find the backscattered current, we use the Keldysh
technique.19 We assume that at t = −⬁, there is no backscattering in the Hamiltonian 关U共x兲 = 0兴, and then the backscattering
is gradually turned on. Thus, at t = −⬁, the numbers NL and
NR of the left- and right-moving electrons are conserved
separately: The system can be described by a partition function with two chemical potentials E1 = EF + eV and E2 = EF
conjugated with the particle numbers NR and NL. This initial
共15兲
共const兲U共n1,m1 ;E = eV兲U共n2,m2 ;eV兲 ¯ U共n p,m p ;eV兲.
共16兲
Such a perturbative expansion can be used as long as
U共n,m;E = eV兲 ⬍ EF
共17兲
for every 共n , m兲. This condition defines the RG cutoff voltage
V* such that
U共n0,m0 ;E = eV*兲 = EF
共18兲
for the most relevant operator U共n0 , m0兲. The RG procedure
cannot be continued to lower-energy scales E ⬍ V*.
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One expects that the leading contribution to the backscattering current emerges in the second order in U共n , m兲 if
the above condition is satisfied. The leading contribution to
the rectification current may, however, emerge in the third
order. Indeed, the second order contributions to the charge
current were computed in Ref. 14. The spin current can be
found in exactly the same way. The result is
bs共2兲
共V兲 ⬃ 兺 共const兲兩U共n,m兲兩2兩V兩2z共n,m兲+1 sgn共V兲. 共19兲
Ic,s
If the 共unrenormalized兲 U共n , m兲 were independent of the
voltage, the above current would be an odd function of the
bias and hence would not contribute to the rectification current. The backscattering amplitudes depend14 on the charge
densities kF␯␴ though, which, in turn, depend on the voltage
in our model.4 The voltage-dependent corrections to the amplitudes are linear in the voltage at low bias eV Ⰶ EF. Hence,
the second order contributions to the rectification currents
scale as U2兩V兩2z共n,m兲+2 共see Appendix B兲. The additional factor of V makes the second order contribution smaller than the
leading third order contribution 关Eq. 共23兲兴 at sufficiently high
impurity strength U Ⰶ EF 共as shown in Appendix B, U / EF
must exceed 关V / EF兴1+z共1,0兲−z共0,1兲−z共1,−1兲兲. Note that the second
order contribution to the rectification current is nonzero even
for a symmetric potential U共x兲 and emerges solely due to the
voltage dependence of the injected charge density 共cf. Sec.
II兲. The leading third order contribution emerges solely due
to the asymmetry of the scatterer.
The main third order contribution comes from the three
backscattering operators most relevant in the renormalization
group sense 关small z共n , m兲, Eq. 共15兲兴. They are identified in
Appendix B. Under conditions 共B6兲–共B8兲, 共B12兲, 共B14兲, and
共B15兲, the most relevant operator is U共1 , 0兲, the second most
relevant U共0 , −1兲, and the third most relevant U共−1 , 1兲. The
cutoff voltage V* is determined by the scaling dimension
z共1 , 0兲, eV* ⬃ EF共U / EF兲1/关1−A兴. The leading nonzero third order contributions to the spin and charge currents come from
the product of the above three operators in the Keldysh perturbation theory 共see Appendix B兲. This leads to
bs
⬃ U3V2共A+B−C−1兲 .
Ic,s
共20兲
This contribution dominates the spin rectification current at
EF共U/EF兲1/关2+2C−2B兴 ⬅ eV** ⬎ eV ⬎ eV* ,
共21兲
as is clear from the comparison with the leading second order
bs
⬃ U2V2A, Eq. 共B10兲. Interestingly, the curcontribution Is,2
rent 关Eq. 共20兲兴 grows as the voltage decreases in the regime
共B6兲–共B8兲, 共B12兲, 共B14兲, and 共B15兲.
However, does the current 关Eq. 共20兲兴 actually contribute
to the rectification effect? In general, Eq. 共20兲 is the sum of
odd and even functions of the voltage and only the even part
is important for us. One might naively expect that such a
contribution has the same order of magnitude for the spin
and charge currents. A direct calculation shows, however,
that this is not the case and the spin rectification current is
much greater than the charge rectification current.
In order to calculate the prefactors in the right hand side
of Eq. 共20兲, one has to employ the Keldysh formalism.
The details are explained in Appendix C. Here, let
us shortly summarize the essential steps: The third order
Keldysh contribution reduces to the integral of
P共t1 , t2 , t3兲 = 具Tc exp(i␾↑共t1兲 + ieVt1 / ប)exp(−i␾↓共t2兲 − ieVt2 / ប)
⫻exp(i关−␾↑共t3兲 + ␾↓共t3兲兴)典 over 共t1 − t3兲 and 共t2 − t3兲, where Tc
denotes time ordering along the Keldysh contour −⬁ → 0 →
−⬁ and the angular brackets denote the average with respect
to the ground state of the noninteracting Hamiltonian 关Eq.
共7兲兴. The integration can be performed analytically as discussed in Appendix C. One finds
Ibs
c =
冏 冏
16e␶2c
eV␶c
sgn共eV兲
␲ប3
ប
a+b+c−2
⌫共1 − a兲⌫共1 − b兲
⫻⌫共2 − a − b − c兲⌫共a + b − 1兲sin
⫻sin
␲a
␲b
sin
2
2
␲共a + b兲
sin ␲共a + b + c兲
2
⫻Re关U共1,0兲U共− 1,1兲U共0,− 1兲兴,
Isbs =
冏 冏
16␶2c
␲a
␲b eV␶c
sin
sin
␲ប2
2
2
ប
共22兲
a+b+c−2
⌫共a + b − 1兲
⫻⌫共2 − a − b − c兲⌫共1 − a兲⌫共1 − b兲
再
⫻ Im关U共1,0兲U共− 1,1兲U共0,− 1兲兴cos
冋
⫻ sin
+ sin
+
␲共a + b + c兲
2
␲共a − b兲
␲共a + b + c兲
␲c
+ cos
sin
2
2
2
␲共a + b兲
␲共a + b + c兲
cos
2
2
册
1
␲共a − b兲
Re关U共1,0兲U共− 1,1兲U共0,− 1兲兴sin
2
2
冎
⫻sin ␲共a + b + c兲sgn共eV兲 ,
共23兲
where a = 2A − 2C, b = 2B − 2C, c = 2C, and ␶c ⬃ ប / EF is the
ultraviolet cutoff time. The charge current 关Eq. 共22兲兴 is an
odd function of the voltage and hence does not contribute to
the rectification effect. The spin current 关Eq. 共23兲兴 is a sum of
an even and odd functions and hence determines the spin
rectification current
Isr =
冏 冏
16␶2c
␲共a + b + c兲 eV␶c
␲a
␲b
sin
sin
cos
2
␲ប2
2
2
ប
a+b+c−2
⫻⌫共a + b − 1兲⌫共2 − a − b − c兲⌫共1 − a兲⌫共1 − b兲
⫻Im关U共1,0兲U共− 1,1兲U共0,− 1兲兴
冋
⫻ cos
+ sin
␲共a − b兲
␲共a + b + c兲
sin
2
2
册
␲共a + b兲
␲共a + b + c兲
␲c
+ sin
cos
.
2
2
2
共24兲
It is nonzero if Im关U共1 , 0兲U共−1 , 1兲U共0 , −1兲兴 ⫽ 0, which is
satisfied for asymmetric potentials. The leading contribution
to the charge rectification currents comes from other terms in
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SPIN CURRENT AND RECTIFICATION IN ONE-…
Isr
V∗
V ∗∗
V
FIG. 5. Qualitative representation of the spin rectification current. The spin current exceeds the charge current and follows a
power-law dependence on the voltage with a negative exponent in
the interval of voltages V* ⬍ V ⬍ V**.
the perturbation expansion. Thus, we expect that in the region 共B6兲–共B8兲, 共B12兲, 共B14兲, and 共B15兲, the spin rectification current exceeds the charge rectification current in an
appropriate interval of voltages 关Eq. 共21兲兴. The difference
between the spin and charge rectification currents can be
easily understood from the limit A = B. In that case, the
charge current changes its sign under the transformation
U共1 , 0兲 ↔ U共0 , −1兲, V → −V. Since U共1 , 0兲 and U共0 , −1兲 enter the current only in the combination U共1 , 0兲U共0 , −1兲, this
means that the charge current must be an odd function of the
voltage bias. A similar argument shows that at A = B, the spin
rectification current is an even function of the voltage in
agreement with Eq. 共23兲.
The voltage dependence of the spin rectification current is
illustrated in Fig. 5. Expression 共20兲 describes the current in
the voltage interval V** ⬎ V ⬎ V*. In this interval, the current
increases as the voltage decreases in the regime 共B6兲–共B8兲,
共B12兲, 共B14兲, and 共B15兲. At lower voltages, the perturbation
theory breaks down. The current must decrease as the voltage decreases below V* and eventually reach 0 at V = 0. At
higher voltages, EF ⬎ eV ⬎ eV**, the second order rectification current 关Eq. 共19兲兴 dominates. The leading second order
contribution Isr ⬃ 兩U共1 , 0兲兩2V2z共1,0兲+2 grows as the voltage increases. The charge rectification current has the same order
of magnitude as the spin current.
The Tomonaga-Luttinger model cannot be used for the
highest voltage region EF ⬃ eV. It is easier to detect charge
currents than spin currents. However, the measurement of the
spin current can be reduced to the measurement of charge
currents: Let us split the right end of the wire into two
branches and place them in opposite strong magnetic fields
so that only electrons with one spin orientation can propagate
in each branch. If both branches are grounded, they still inject exactly the same charge and spin currents into the wire
as one unpolarized lead. However, the current generated in
the wire will split between two branches into the currents of
spin-up and spin-down electrons. If they are opposite, then
pure spin current is generated.
V. CONCLUSIONS
In this paper, we have shown that rectification in quantum
wires in a uniform magnetic field can lead to a spin current
that largely exceeds the charge current. The paper focuses on
the regime of low voltages and weak asymmetric potentials
in which the perturbation theory provides quantitatively exact predictions. Qualitatively, the same behavior is expected
up to eV , U ⬃ EF. The spin rectification effect is solely due to
the properties of the wire and does not require timedependent magnetic fields or spin-polarized injection as from
magnetic electrodes. The currents are driven by the voltage
source only. In an interval of low voltages, the spin current
grows as the voltage decreases. In contrast to some other
situations, the z component of the total spin is conserved and
hence the dc spin current is constant throughout the system.
ACKNOWLEDGMENTS
We thank J. B. Marston and D. Zumbühl for many helpful
discussions. This work was supported in part by the NSF
under Grants No. DMR-0213818, No. DMR-0544116, and
No. PHY99-07949 and by the Salomon Research Award.
D.E.F. acknowledges the hospitality of the Aspen Center for
Physics, of the MPI Dresden, and of the KITP Santa Barbara
where this work was completed.
APPENDIX A: HIGH POTENTIAL BARRIER
In this appendix, we first briefly consider the model of
noninteracting electrons, Sec. II, and then a simple Hartreetype model for strongly interacting electrons.
1. Model without interaction
We consider noninteracting electrons in the presence of
the potential
U共x兲 = u1␦共x兲 + u2␦共x − a兲.
共A1兲
The transmission coefficient can be found from elementary
quantum mechanics,
T共E兲 =
1
,
共1 − 2s1s2 sin ka兲 + 共s1 + s2 + s1s2 sin 2ka兲2
2
2
共A2兲
where E = ប2k2 / 2m and si = mui / kប2. The spin and charge
rectification currents can be computed from Eqs. 共1兲 and 共2兲.
Figure 3 shows their voltage dependence for a certain choice
of u1, u2, the voltage bias V, and the magnetic field H.
2. Model with interaction
It is difficult to find a general analytic expression for the
current in the regime when both the electron interaction and
potential barrier are strong. If all characteristic energies, U,
eV, ប2 / 关ma2兴, and the typical potential energy of an electron
E P, are of the order of EF, then one can estimate the spin and
charge rectification currents with dimensional analysis: Irc
⬃ eEF / ប, Isr ⬃ EF.
To obtain a qualitative picture of the interaction effects in
the case of a high potential barrier 关Eq. 共A1兲兴, we restrict our
discussion to a simple model in the spirit of the zero-mode
approximation.20 We assume that electrons move in a selfconsistent Hartree-type field. In our ansatz, the selfconsistent field takes three different constant values VL, V M ,
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PHYSICAL REVIEW B 76, 085119 共2007兲
BRAUNECKER, FELDMAN, AND LI
and VR on the left of the potential barrier, between two
␦-function scatterers, and on the right of the potential barrier.
In the spirit of the Luttinger liquid model, we assume that the
constants VL, V M , and VR are proportional to the average
charge density in the respective regions, e.g., V M
= ␥a 兰a0dx␳共x兲, where ␥ is the interaction constant.
A result is shown in Fig. 4. We see that the voltage dependence of the spin and charge rectification currents exhibits a behavior similar to that of the noninteracting case.
APPENDIX B: ESTIMATION OF HIGHER PERTURBATIVE
ORDERS
In this appendix, we compare contributions to the rectification currents from different orders of perturbation theory.
We focus on the regime when the third order contribution
dominates. This appendix contains five sections and has the
following structure: 共1兲 We introduce a parametrization for
the scaling dimensions 关Eq. 共15兲兴. 共2兲 We discuss the operators most relevant in the RG sense. 共3兲 We determine at what
conditions the second order contribution to the rectification
current dominates. Appendix B 3 also contains a lemma
which is important in Appendix B 4. 共4兲 We determine at
what conditions the third order contribution to the current
dominates. 共5兲 We estimate the voltages and currents at
which the spin rectification current can exceed the charge
rectification current in realistic systems.
As shown in Refs. 4 and 5, there are two effects leading to
rectification in Luttinger liquids, which are here very shortly
summarized: the density-driven and the asymmetry-driven
rectification effects. The former appears at second order in
U. It appears because the backscattering potential depends on
the particle densities in the system, which, in turn, are modified by the external voltage bias. The leading order backscattering currents are of the form14 Ibs共V兲 ⬃ sgn共V兲U2兩V兩␣,
so that the rectification currents Ir = 关Ibs共V兲 + Ibs共−V兲兴 vanish.
Due to the density dependence, however, an expansion of U
to linear order in V cancels the sgn共V兲, and we obtain a
rectification current Ir ⬃ U2兩V兩␣+1.
The asymmetry-driven rectification effect appears at third
order in U. It is due solely to the spatial asymmetry of the
potential U共x兲: Due to backscattering off U, screening
charges accumulate close to the impurity. These create an
electrostatic nonequilibrium backscattering potential W共x兲
for incident particles, leading to an effective potential Ū共x兲
= U共x兲 + W共x兲. The spatial distribution of charges follows
from the shape of U共x兲 and the applied voltage bias. An
asymmetric U共x兲 leads to different electrostatic potentials for
positive or negative bias and hence to rectification. If we
expand the current, Ibs ⬃ Ū2 ⬃ U2 + UW + ¯, the asymmetry
appears first at order UW. Since the charge density in the
vicinity of the impurity is modified by the modification of
the particle current through backscattering, W itself is 共selfconsistently兲 related to the backscattering current as W ⬃ Ibs.
Hence, W ⬃ U2, so that the asymmetric rectification effect
appears first at third order in U, Ir ⬃ UW ⬃ U3.
The main result of this paper are expressions for the currents that result from the perturbation theory at third order in
the impurity potential U. In this appendix, we show that the
considered contribution indeed dominates the second and
other third order expressions in the region defined by Eqs.
共B6兲–共B8兲, 共B12兲, 共B14兲, and 共B15兲. In addition, we give the
proof that higher perturbative orders N 艌 4 cannot exceed
these values in the considered range of the system parameters gc, gs, ␣, and ␤. Unless we want to emphasize the
correct dimensions, we set EF = 1, e = 1, and ប = 1 in this appendix. We assume that U ⬍ EF and eV ⬍ EF.
An important observation is the following: In Eq. 共16兲,
兺ini = 兺imi = 0. This follows from the fact that in the absence
of backscattering, the numbers of right and left movers with
different spin orientations are conserved.
1. Parametrization of scaling dimensions
According to Eq. 共15兲,
z共n,m兲 = n2A + m2B + 2nmC − 1,
共B1兲
A = 关gc共1 + ␣兲2 + gs共1 + ␤兲2兴,
共B2兲
B = 关gc共1 − ␣兲2 + gs共1 − ␤兲2兴,
共B3兲
C = 关gc共1 − ␣2兲 − gs共1 − ␤2兲兴.
共B4兲
where
Since gc and gs are positive, A and B are also positive. C can
have any sign. It satisfies the inequality
兩C兩 ⬍ 冑AB.
共B5兲
Indeed, AB − C = 4gcgs共1 − ␣␤兲 ⬎ 0. Any values of A , B ⬎ 0
and −冑AB ⬍ C ⬍ 冑AB are possible. For example, one can set
␣ = ␤ = 共冑A − 冑B兲 / 共冑A + 冑B兲, gc = 共冑A + 冑B兲2关1 + C / 冑AB兴 / 8,
and gs = 共冑A + 冑B兲2关1 − C / 冑AB兴 / 8.
2
2
2. Most relevant operators
Depending on the values of A, B, and C, many different
possibilities for relative importance of different backscattering operators U共m , n兲 exist. In this paper, we focus on the
situation when the most relevant operator is U共1 , 0兲, the second most relevant operator is U共0 , −1兲, and the third most
relevant operator is U共−1 , 1兲 关certainly, the scaling dimensions of the operators U共n , m兲 and U共−n , −m兲 are always the
same兴. We will also assume that the operator U共1 , 0兲 is relevant in the RG sense, i.e., z共1 , 0兲 ⬍ 0. The analysis of the
situation in which U共0 , −1兲 is the most relevant operator,
U共1 , 0兲 is the second most relevant, and U共−1 , 1兲 is the third
most relevant follows exactly the same lines. Similarly, little
changes if U共1 , 1兲 is the third most relevant operator.
The scaling dimensions of the three aforementioned operators are A − 1, B − 1, and A + B − 2C − 1. The following inequality must be satisfied in order for these operators to be
most relevant backscattering operators: A − 1 ⬍ B − 1 ⬍ A + B
− 2C − 1⬍ 共all other scaling dimensions兲. Hence,
B ⬎ A ⬎ 2C.
Since z共1 , 0兲 ⬍ 0,
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PHYSICAL REVIEW B 76, 085119 共2007兲
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A ⬍ 1.
共B7兲
When are all other operators less relevant? We must consider
three classes of operators: 共1兲 U共1 , 1兲, 共2兲 U共n , 0兲 and
U共0 , n兲 with 兩n兩 ⬎ 1, and 共3兲 all other operators.
共1兲 Since z共1 , 1兲 = A + B + 2C − 1, one finds
C ⬎ 0.
contains U共±1 , 0兲 and U共0 , ± 1兲 only, then it must contain
VU1 as discussed above兴. Thus, any rectification current contribution with Ũ only cannot exceed U3V2z共1,0兲+z共1,−1兲+1. Comparison with Eq. 共B10兲 at V ⬃ V* leads to the condition
B ⬎ 2C + 1.
共B11兲
共B8兲
共2兲
z共0 , n兲 = Bn2 − 1 ⬎ z共n , 0兲 = An2 − 1 艌 4A − 1 ⬎ A + B
− 2C − 1. Thus,
3A + 2C ⬎ B.
共B9兲
共3兲
z共n , m兲 − 共A + B − 2C − 1兲 = An + Bm + 2Cnm − 共A + B
− 2C兲 艌 An2 + Bm2 − C共n2 + m2兲 − A − B + 2C = 共A − C兲共n2 − 1兲
+ 共B − C兲共m2 − 1兲 ⬎ 0, since B − C ⬎ A − C ⬎ 0 in accordance
with Eq. 共B6兲, 兩n兩 , 兩m兩 艌 1, and either 兩n兩 or 兩m兩 exceeds 1.
Thus, case 共3兲 gives no new restriction on A, B, and C.
2
2
4. Third order contribution to the current
The most interesting question is different. When does the
third order contribution dominate the rectification current?
We will focus on the third order contribution I3 proportional
to U共1 , 0兲U共0 , −1兲U共−1 , 1兲 at V ⬃ V*. Note that this contribution is proportional to ⬃V2共A+B−C−1兲 and hence scales as a
negative power of the voltage if
A + B ⬍ C + 1.
At V ⬃ V , U ⬃ V
*
When is the second order contribution to the rectification
current dominant? Any operator U共n , m兲 can be represented
as Ũ共n , m兲 + VU1共n , m兲 + ¯, where U1 ⬃ Ũ / EF. Any second
order contribution to the current which contains Ũ only is an
odd function of the voltage bias. Indeed, any such contribution is proportional to Ũ共n , m兲Ũ*共n , m兲 = Ũ共n , m兲Ũ共−n , −m兲.
The transformation Ũ ↔ Ũ*, V → −V changes the sign of the
current. At the same time, the transformation Ũ ↔ Ũ* cannot
change the second order current at all. Hence, it is odd in the
voltage. The same argument applies to any perturbative contribution which contains only Ũ, if every operator Ũ共n , m兲
enters in the same power as Ũ*共n , m兲. In particular, if only
two operators Ũ共n , 0兲 and Ũ共0 , m兲 and their conjugate enter
then the resulting current, the contribution is odd.
Thus, all second order contributions to the rectification
current must contain U1. As is clear from Eq. 共19兲, the leading second order contribution is proportional to the square of
the most relevant operator, 兩U共1 , 0兲兩2. It scales as
共B10兲
In this section, we discuss at what conditions this contribution dominates for all V ⬎ V* 关see Eq. 共18兲兴. Since U共1 , 0兲
is the most relevant operator, its renormalized amplitude
U共1 , 0 ; E = V兲 exceeds the renormalized amplitude of all
other operators on every energy scale. At the same time, it
remains lower than 1 共i.e., EF兲 for V ⬎ V*. This certainly
means that the renormalized amplitudes are smaller than 1
for all other operators too. Hence, the product of any operators is smaller than the product of any two of them and that
product cannot exceed U2共1 , 0 ; E兲. This guarantees that the
second order current 关Eq. 共B10兲兴 exceeds any second or
higher order contribution which contains any operator
VU1共n , m兲. Thus, we have to compare I2 with higher order
contributions to the rectification current which contain Ũ
only. Every such contribution is at least third order and contains at least one operator less relevant than U共0 , −1兲 关if it
共B12兲
. Thus,
I3共V = V*兲 ⬃ V2B−A−2C+1 .
3. Second order contribution to the current
I2 ⬃ VU2V2z共1,0兲+1 ⬃ U2V2A .
1−A
共B13兲
We need to compare I3, Eq. 共B13兲, with the following types
of contributions: 共1兲 those containing at least three different
operators 关we treat a pair of U共n , m兲 and U共−n , −m兲
= U*共n , m兲 as one operator兴, 共2兲 those containing only one
type of operators, and 共3兲 those containing two types of operators.
Cases 共1兲 and 共2兲 are easy.
共1兲 I3 contains the product of the three most relevant operators and hence always exceeds the product of any other
three different operators at any energy scale EF ⬎ V ⬎ V*.
Any contribution with three different operators is the product
of three different operators times perhaps some other combination of operators which cannot exceed 1 at EF ⬎ V ⬎ V*.
Hence, it is smaller than I3.
共2兲 Any contribution to the rectification current with only
one type of operators must contain VU1. As discussed in the
previous section, the leading contribution of such type
emerges in the second order. It is I2, Eq. 共B10兲. At V ⬃ V*,
I2共V = V*兲 ⬃ V2. The condition I2共V*兲 ⬍ I3共V*兲 means that
2B ⬍ A + 2C + 1.
共B14兲
共3兲 We have to consider three possibilities: 共3.1兲 one operator has the form U共n , 0兲 and the second operator has the
form U共k , m兲, m ⫽ 0 or one operator has the form U共0 , m兲
and the other one has the form U共n , k兲, n ⫽ 0; 共3.2兲 both
operators have the form U共ni , 0兲 or both operators have the
form U共0 , mi兲; and 共3.3兲 both operators have the form
U共ni , mi兲 with ni, mi ⫽ 0.
共3.1兲. Let us assume that one operator has the form
U共n , 0兲 and the second one is U共k , m兲. The case of the operators U共0 , m兲 and U共n , k兲 can be considered in exactly the
same way. We must have the same number of operators
U共k , m兲 and U共−k , −m兲 in the perturbative contribution since
the sum of the second indices ±m must be 0. 关The other cases
are covered in 共3.3兲.兴 From the analysis of the sum of the first
indices, one concludes that the operators U共n , 0兲 and
U共−n , 0兲 also enter in the same power. It follows from the
previous section that the perturbative contribution must con-
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PHYSICAL REVIEW B 76, 085119 共2007兲
BRAUNECKER, FELDMAN, AND LI
tain at least one U1 operator and hence is smaller than I2.
Hence, it is also smaller than I3.
共3.2兲 We will focus on the case when both operators have
the form U共ni , 0兲. The case when both operators have the
form U共0 , mi兲 is very similar and does not lead to a new
restriction on A, B, and C. The scaling dimensions of the
operators U共n , 0兲 are An2 − 1. Operators with greater n are
less relevant. Since the contribution contains two different
operators, it must be at least third order 关we treat U共n , 0兲 and
U共−n , 0兲 as the same operator兴. At least one of the two operators must have 兩ni兩 ⬎ 1 关otherwise, all operators are
U共±1 , 0兲兴. Thus, the contribution cannot exceed U2共1 , 0 ; E
= V兲U共2 , 0 ; E = V兲 ⬃ U3V6A−2. The comparison with I3
⬃ U3V2A+2B−2C−2 at EF ⬎ V ⬎ V* yields
B ⬍ 2A + C.
APPENDIX C: EXPLICIT EVALUATION OF THE THIRD
ORDER CURRENTS
The charge or spin currents in the third order in the potentials U are evaluated from the following perturbative expression:
bs共3兲
Ic,s
共V兲 =
5. Numerical estimates
In order to get a feeling about the magnitude of the effect,
let us consider a particular choice of parameters A = B
= 7 / 12, C = 7 / 24, eV ⬃ 0.01EF, and eV* ⬃ 10−4EF. For such
A, B, and C, the scaling dimensions of the three most relevant operators are the same. The inequalities 共B7兲, 共B12兲,
共B14兲, and 共B15兲 are satisfied. The equality A = B = 2C corresponds to a limiting case of Eq. 共B6兲. One finds that U
⬃ 0.01EF and eV** ⬃ 0.1EF. Repeating the arguments of the
previous section, one can estimate the leading correction to
I3 as ␦I ⬃ 共eV / EF兲7/12I3 Ⰶ I3. The spin rectification current is
the difference of two opposite electric currents of the spin-up
and -down electrons times ប / 关2e兴. Even if EF is as low as
⬃0.1 meV, this still corresponds to the voltage V of the order of microvolts and the currents21 关Eq. 共24兲兴 of spin-up and
-down electrons of the order of picoamperes, i.e., within the
ranges probed in experiments with semiconductor heterostructures. Certainly, the current increases if EF or V* is increased.
兺 共n↑ ± n↓兲
冕
CK
dt1dt2
具TcÛ共n↑,n↓ ;0兲
ប2
⫻Û共m↑,m↓ ;t1兲Û共l↑,l↓ ;t2兲典,
共C1兲
where the sum runs over indices satisfying n␴ + m␴ + l␴ = 0 for
␴ = ↑ , ↓, CK is the Keldysh contour −⬁ → 0 → ⬁, Tc is the
time order on CK, and we omitted a constant prefactor. The
operators Û are given by
共B15兲
Note that the above condition is stronger than Eq. 共B9兲.
共3.3兲 This case is easy: the contribution must be at least
third order again. Both operators U共ni , mi兲 are less relevant
than U共1 , 0兲 and U共0 , −1兲 and no more relevant than U共1 ,
−1兲. Thus, the contribution is automatically smaller than I3 at
any energy scale EF ⬎ V ⬎ V*.
We now have a full set of conditions at which the third
order contribution dominates at V ⬃ V* and the spin rectification current scales as a negative power of the voltage.
These are Eqs. 共B6兲–共B8兲, 共B12兲, 共B14兲, and 共B15兲.
The above analysis shows that I3 exceeds any contribution
to the spin rectification current which does not contain VU1
in the whole region EF ⬎ V ⬎ V*. I2 dominates the remaining
contributions for any V ⬎ V*. The contributions become
equal, I2 = I3, at V = V** = U1/关2+2C−2B兴. In the interval of voltages V** ⬎ V ⬎ V*, the spin rectification current is dominated
by I3. At V ⬎ V**, the spin and charge rectification currents
are dominated by I2.
共− i兲3
2!
Û共n↑,n↓ ;t兲 = U共n↑,n↓兲ei共n↑+n↓兲teV/បein↑␾↑共t兲+in↓␾↓共t兲 . 共C2兲
The most relevant expressions are those arising
from the combinations U共1 , 0兲U共0 , −1兲U共−1 , 1兲 and
U共−1 , 0兲U共0 , 1兲U共1 , −1兲 共see Appendix B兲. The third order
contributions to the current contain correlation functions of
the form
P共t1,t2,t3兲 = 具Tce±i关␾↑共t1兲−␾↓共t2兲−␾↑共t3兲+␾↓共t3兲兴典.
共C3兲
We evaluate the correlation functions within the quadratic
model described by Eq. 共7兲 and use relations 共8兲 and
˜ c,s共t1兲␾
˜ c,s共t1兲典 = −2 ln(i共t1 − t2兲 / ␶c + ␦), with an infinitesimal
具␾
␦ ⬎ 0 and ␶c ⬃ ប / EF the ultraviolet cutoff time. This leads to
P共t1,t2,t3兲 = 关iTc共t1 − t3兲/␶c + ␦兴2C−2A关iTc共t2 − t3兲/␶c + ␦兴2C−2B
⫻关iTc共t1 − t2兲/␶c + ␦兴−2C ,
共C4兲
where Tc共ti − t j兲 = 共ti − t j兲, if time ti stays later than t j on the
Keldysh contour, and otherwise, Tc共ti − t j兲 = 共t j − ti兲. Expression 共C4兲 is independent of the ⫾ signs in Eq. 共C3兲.
The spin and charge current contributions, proportional to
U共1 , 0兲U共0 , −1兲U共−1 , 1兲, are complex conjugate to those
proportional to U共−1 , 0兲U共0 , 1兲U共1 , −1兲 = U*共1 , 0兲U*共0 ,
−1兲U*共−1 , 1兲. Thus, it is sufficient to calculate only the contributions of the first type. In the case of the charge current,
their calculation reduces to the calculation of the following
two integrals over the Keldysh contour:
冕
dt1dt3 P共t1,0,t3兲exp共ieVt1/ប兲
共C5兲
dt2dt3 P共0,t2,t3兲exp共− ieVt2/ប兲.
共C6兲
and
冕
One of the times t1 and t2 is zero since the current operator is
taken at t = 0 in Eq. 共13兲. The two integrals can be evaluated
in exactly the same way. We will consider only the first
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PHYSICAL REVIEW B 76, 085119 共2007兲
SPIN CURRENT AND RECTIFICATION IN ONE-…
integral. We find eight integration regions. They correspond
to 2 ⫻ 2 = 4 possibilities for the branches of the Keldysh contour on which t1 and t3 are located and two possible relations
兩t1兩 ⬎ 兩t3兩 or 兩t3兩 ⬎ 兩t1兩. In all eight cases, we first integrate over
t3. The integral reduces to the Euler B function. Then, we
integrate over t1. This yields a ⌫ function. Finally, we obtain
Eq. 共22兲.
The spin current contains three contributions proportional
to U共1 , 0兲U共0 , −1兲U共−1 , 1兲. Two of them reduce to the integrals 共C5兲 and 共C6兲. The third contribution is proportional to
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冕
dt1dt2 P共t1,t2,0兲exp共ieV关t1 − t2兴/ប兲.
共C7兲
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